Persistent homology can be differentiated and incorporated into learning pipelines, but no analogous framework exists for zigzag persistence, which is needed when the underlying topological structure evolves non-monotonically over time. We develop such a framework for sequences of simplicial complexes obtained by thresholding time-dependent filtering values on a fixed complex. By assigning persistence diagram endpoints the real-valued times at which linearly interpolated filtering values cross the threshold, we transfer the continuity of the filtering values to the diagram points. This yields smooth local lifts of the resulting persistence-diagram-valued map, from which we derive differentials almost everywhere under mild regularity conditions on the parametrization of the filtering values. We prove local Lipschitz continuity outside an explicit measure-zero exclusion set; standard stochastic subgradient convergence guarantees therefore do not apply directly. We argue that, even without such guarantees, this exclusion set is small enough in practice to allow effective optimization. We test this empirically in two experiments: sensor network coverage optimization and dynamic graph classification.
Figures & tables
Figure 1: Commutative diagrams of the constructions of Corollaries 3.3 ( 1(a) ) and 3.7 ( 1(b) ), representing the function Pp∣U . The continuous line denotes the maps whose differentials we compute and compose to obtain the differential of the local lift B .
Figure 2: The experiment. Top: four snapshots of the lifetime strategy’s final iterate on the median seed 18 ; t is elapsed time. Cells are shaded by how long they have waited since last covered, pale to red past Δ=6 ; black squares are the fixed sensors, coloured dots the robots, with sensing range dashed and the last three frames of motion trailing. Bottom left: the same for all nine cells over the 30 frames, snapshots dotted; the streaks keep resetting because every cell keeps being revisited. Right: the total objective L along the iterate sequence, penalties included, held-out seeds 10 – 19 , median bold and seeds faint, both axes logarithmic; the rise at k=2 is the first perturbation breaking the unit-speed penalty.
Figure 3: The experiment. Each example is two figure-eights whose square nodes switch so that one loop of each is closed at any time; how fast the signal alternates is the label, and the decoy alternates at the opposite rate. The node features go through the depicted pipeline, which predicts whether the signal alternates slowly or rapidly.
Appendix figures & tables14 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 4: Example C.18 : an interval broken in the middle ( Npϵ+ ). In each panel the upper plot is the piecewise-linear interpolation of the filtering values v at i=0,…,N+1 , with the threshold ϵ dashed, and the lower plot is the resulting H0 barcode on the same time axis. In (a) the value at i=2 equals ϵ with both neighbours above it, so the exit and the re-entry occur at the same time and cancel as a phantom pair, leaving a single bar. In (b) an arbitrarily small δ>0 makes both crossings genuine, and the bar breaks in two.
Figure 5: Example C.19 : an interval of non-vanishing length created from nothing ( Np,p+1ϵ,ϵ ). Same conventions as Figure 4 . In (a) two consecutive values equal ϵ , so σ never crosses the threshold and the diagram is empty. In (b) raising the first of them by δ>0 leaves the value at i=2 exactly on the threshold, but its right neighbour is now below it, so the configuration is not phantom: the exit is genuine and occurs at t=2 , producing the bar (ϵ+δϵ,2) , whose length tends to 1 and not to 0 . In (c) raising both values instead gives the bar (ϵ+δϵ,3−ϵ+δϵ) and the same conclusion holds.
Figure 6: The set N0 is represented in red as a subset of R3∣K∣≅(R3∣K∣)∗=FF3(K) . The highlighted red lines form N0,1ϵ,ϵ , while the transparent red region represents N0ϵ+ . The complementary set of N0 is open and connected.
Figure 7: The arena. Left: with no robots each of the nine cells is a hole, so the 1 -Betti number β1=9 ; the two circles on one sensor are the Rips radius R=1.05 , which reaches the four side neighbours and not the diagonals, and the sensing radius RS=0.61 ; the pink diamonds are the area no sensor covers, the cell centres lying 0.707 from their corners. Right: three robots fill three cells, β1=6 . Three robots can never close more than a third of the arena at once, which is why the total hole-time is close to a fixed budget and its distribution in time is the quantity at issue.
strategy
Ltopo : init → final
longest bar
area >Δ
count (at Δ=6 )
244.8→211.0
21.6 ( 11.3 – 29.0 )
2.7% ( 2.3 – 3.2 )
lifetime, Δ=6
507.9→33.8
09.4 ( 08.0 – 10.9 )
1.9% ( 0.7 – 2.9 )
lifetime, Δ=8
310.7→17.2
11.1 ( 09.0 – 13.9 )
0.7% ( 0.4 – 1.9 )
lifetime, Δ=10
180.1→06.7
11.8 ( 10.3 – 16.2 )
0.5% ( 0.1 – 1.2 )
lifetime, Δ=12
096.9→05.4
13.8 ( 12.3 – 21.0 )
0.1% ( 0.0 – 0.9 )
Appendix
Table 1: Topological results of the three-robots-experiment, for count and lifetime strategies, the last with four different tolerance Δ values. The count objective carries no Δ , so its waiting area is computed for Δ=6 .
robots
strategy
Ltopo : init → final
longest bar
area >Δ
2
count
250.8→227.9
25.4 ( 19.9 – 29.0 )
3.4% ( 3.1 – 3.6 )
2
lifetime
996.2→144.6
12.4 ( 10.6 – 15.9 )
3.3% ( 2.7 – 3.7 )
3
count
244.8→211.0
21.6 ( 11.3 – 29.0 )
2.7% ( 2.3 – 3.2 )
3
lifetime
507.9→33.8
09.4 ( 08.0 – 10.9 )
1.9% ( 0.7 – 2.9 )
4
count
239.2→197.0
17.1 ( 09.6 – 24.6 )
1.8% ( 1.3 – 2.4 )
4
lifetime
250.4→14.8
08.9 ( 06.7 – 10.9 )
0.5% ( 0.0 – 1.3 )
Appendix
Table 2: Robot-count ladder, lifetime objective at Δ=6 throughout.
strategy
condition
Ltopo : init → final
longest bar
area >Δ
lifetime
gradient and noise (reported)
507.9→33.8
09.4 ( 08.0 – 10.9 )
1.9% ( 0.7 – 2.9 )
lifetime
gradient only, σ=0
507.9→48.7
10.6 ( 07.3 – 29.0 )
2.1% ( 0.7 – 3.4 )
lifetime
motion only, Ltopo removed
507.9→530.9
21.3 ( 18.9 – 29.0 )
2.2% ( 2.0 – 2.8 )
count
motion only, Ltopo removed
244.8→246.7
21.3 ( 18.9 – 29.0 )
2.2% ( 2.0 – 2.8 )
Appendix
Table 3: Three experiments with three robots and Δ=6 : stochastic optimization, optimization without noise (gradient only), stochastic optimization without topological term (motion only). In the motion only rows the topological term is removed from the loss, so both strategies give the same run: each row reads its own term along that path, evaluated but not optimized.
Figure 8: The count strategy along the iterate sequence, held-out seeds 10 – 19 , median bold and seeds faint, logarithmic iteration axis. Left: the count term Ltopo=∑(b,d)∈D(d−b) on a linear value axis; it descends by 14% and no more, because with three robots at least six of the nine cells are open at every instant and the loss has a hard floor. Right: the total loss L , logarithmic value axis; the peak at k=2 is the first perturbation breaking the unit-speed penalty (which was satisfied by the random walk initialization), and the curve ends 1.1 above Ltopo .
Figure 9: Left: the topological term at the iterate for the lifetime objective at each tolerance, medians over held-out seeds 10 – 19 , logarithmic axes. Right: the longest H1 bar at the same iterates, each condition against its own Δ (dashed, matching colours). The objective goes on descending for the whole run at every tolerance.
Figure 10: The total objective L of the lifetime strategy at each tolerance, penalties included, medians over held-out seeds 10 – 19 , logarithmic axes.
Figure 11: The designed task. (a) One figure-eight: two loops sharing a node, each closed while its switch vertex is active. (b) The switch patterns defining the two classes, slow against fast. (c) The resulting Betti curves, identical between the classes at every frame, so a per-frame reading has nothing to separate. (d) The zigzag barcodes of one example of each class under the filtering function the generator knows, at the threshold of the experiment: three bars with the longest at 0.48 against seventeen with the longest at 0.06 .
setting
value
why
γ0
0.02
scanned over {0.01,0.02,0.03,0.05,0.1} ; the optimum is interior, larger steps moving the field further and classifying worse
α
0.6
in (1/2,1] , matching the step-size condition of Proposition B.7 , which on its own brings none of its other hypotheses;
iterations
3200
at 800 the loss is still falling when the schedule has decayed the step to nothing; four times the budget removes that truncation
minibatch
16 , uniform with replacement
the sampling of Definition B.6
ridge λ
10−4
no measurable effect in development
amplification
30
the rescaling of the graph convolution’s weights; without it the graph convolution barely moves beside the reader (Appendix E.1 )
Appendix
Table 4: Stage-2 settings and why they take these values. All were fixed on development draws of the generator, none of which the reported results come from.
read-out
field
runs
solved
accuracy
zigzag
learned
best of 4
20/20
0.995±0.016
single
58/80
0.893±0.178
per-frame persistence
learned
best of 4
0/20
0.755±0.112
single
0/80
0.635±0.136
zigzag
signal-only
reader only
20/20
1.000±0.000
per-frame persistence
signal-only
reader only
0/20
0.500±0.000
Appendix
Table 5: Temporal-graph classification on the same twenty held-out splits; final-iterate test accuracy, chance =0.500 ; solved counts splits above 0.9 . Learned variants: best of four seeds by final validation loss, and all 80 single runs.
Figure 12: Loss of the runs the protocol keeps: on each of the twenty held-out splits, the seed with the lowest final validation loss among four, which solves its split in every case. Faint lines are the individual validation curves; the solid line is their mean, the dashed line the mean training loss. Both axes logarithmic; the dotted line marks ln2 , the loss at chance.
Persistence-based topological optimization deforms a point cloud X⊂Rd by minimizing objectives of the form L(X)=ℓ(Dgm(X)), where Dgm(X) is a persistence diagram. In practice, optimization is limited by two coupled issues: persistent homology is typically computed on subsamples, and the resulting topological gradients are highly sparse, with only a few anchor points receiving nonzero updates. Motivated by diffeomorphic interpolation, which extends sparse gradients to smooth ambient vector fields via Reproducing Kernel Hilbert Space (RKHS) interpolation, we propose a more scalable pipeline that improves both subsampling and gradient extension. We introduce subsampling via random slicing, a lightweight scheme that promotes iteration-wise geometric coverage and mitigates density bias. We further replace the costly kernel solve with a fast Nadaraya-Watson (NW) Gaussian convolution, producing a globally defined smooth update field at a fraction of the computational cost, while being more suited for topological optimization tasks. We provide theoretical guarantees for NW smoothing, including anchor approximation bounds and global Lipschitz estimates. Experiments in 2D and 3D show that combining random slicing with NW smoothing yields consistent speedups and improved objective values over other baselines on common persistence losses.
We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods. In particular, we provide empirical evidence that persistent discrete homology is more noise-resistant than persistent homology of the Vietoris-Rips complex for data coming from non-metric settings.
Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at \href{https://github.com/Aalto-QuML/Hourglass}{this https URL}.
Mattie Ji, Indradyumna Roy, Vikas Garg
University of Pennsylvania · Aalto University · Indian Institute of Technology Bombay +1